Polyhedral metric

The intrinsic metric (cf. Internal metric) of a connected simplicial complex of Euclidean simplices, in which identified boundaries are isometric and identification is carried out by an isometry (cf. Isometric mapping). The distance between two points of a complex is the infimum of the lengths of the polygonal lines joining the points and such that each link is within one of the simplices. An example of a polyhedral metric is the intrinsic metric on the surface of a convex polyhedron in $E^3$. A polyhedral metric can also be considered on a complex of simplices in a space of constant curvature.

Polyhedral metrics allow synthetic methods of research. Usually polyhedral metrics are considered for complexes which are manifolds or manifolds with boundary. In the theory of convex surfaces and two-dimensional manifolds of bounded curvature (cf. Two-dimensional manifold of bounded curvature), approximation by means of polyhedral metrics serves as a universal tool for research (see [1], [2]). In the study of convex surfaces, three-dimensional polyhedral metrics have also been successfully used (see [3]). Certain results of global Riemannian geometry have been generalized to polyhedral metrics of dimension $n>2$ (see , [5]).

References

[1]	A.D. Aleksandrov, "Die innere Geometrie der konvexen Flächen" , Akademie Verlag (1955) (Translated from Russian)
[2]	A.D. Aleksandrov, V.A. Zalgaller, "Two-dimensional manifolds of bounded curvature" Trudy Mat. Inst. Steklov. , 63 (1962) (In Russian)
[3]	Yu.A. Volkov, "Existence of a convex polyhedron with given curvature I" Vestnik Leningrad. Univ. , 15 : 19 (1960) pp. 75–86 (In Russian) (English abstract)
[4a]	A.D. Milka, "Multidimensional spaces with polyhedral metric of nonnegative curvature I" Ukrain. Geom. Sb. , 5–6 (1968) pp. 103–114 (In Russian)
[4b]	A.D. Milka, "Multidimensional spaces with polyhedral metric of nonnegative curvature II" Ukrain. Geom. Sb. , 7 (1970) pp. 68–77, 185 (In Russian)
[5]	D.A. Stone, "Geodesics in piecewise linear manifolds" Trans. Amer. Math. Soc. , 215 (1976) pp. 1–44
[a1]	H. Busemann, "Convex surfaces" , Interscience (1958) pp. Chapt. IV